Number Bases (Radices)

In mathematical numeral systems, the radix or base is the number of unique digits, including zero, used to represent numbers in a positional numeral system. For example, for the decimal system (the most common system in use today) the radix is ten, because it uses the ten digits from 0 through 9. Source

Today I turned 24793 days old, a prime number of days, and as it turns out a palindrome in base 12 (12421). I started playing around with the representation of 24793 in other number bases or radices. In base 31, the representation is poo so I may be in for a shitty day. In base 36, the letters of the alphabet are exhausted:

Beyond base 36, the following system is used:

More generally, in a system with radix b (b > 1), a string of digits \(\ d_1 … d_n \) denotes the number \(\ d_1b^{n−1}+d_2b^{n−2}+ … +d_nb^{0} \), where \(\ 0 ≤ d_i < b \).

In practice, a colon is used to separate the individual digits e.g. 24793 is written\(\ 18:4:3 \,_{37} \)  in base 37 which can be dispensed with of course in the case of base 10.

Radices are usually natural numbers but they don't have to be. For example, a base using the golden ratio\(\ \Phi\) is possible, remembering that\(\ \Phi\) is given by the expression: \[\ \frac{1+\sqrt{5}}{2}\] Commonly, the capital letter\(\ \Phi\) is used to represent 1.618033988749895… and the lower case letter \( \phi\) to represent 0.618033988749895… or\(\ \Phi-1\) but this is certainly not always the case.

Such a base is colloquially called phinary and the following table gives some idea of how it works (source) but uses\(\ \varphi\), a variation of the lowercase\(\ \phi\) but meant to equal 1.618033988749895… here:

\(\ \Phi\) is closely linked to the Fibonacci sequence since \[ \lim_{n \to \infty} \frac{F_n}{F_{n-1}}=\Phi \]More information about \( \Phi  \) as a number base can be found on this site.